TFPT scientific source excerpt Original repository path: experiments/theory-contracts/recursive-mirrored-source-20261103/POSITIVE-GRAPH-MEASURE-TEST.txt Original file SHA-256: 0a4f52f2081c324c7b144213f3e5880328866764781a043b813c52b322ea9232 Inclusive original lines: 161-269 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000161 | In the unbounded circle source, R itself can be unbounded although the 000162 | given G(h),C,X are bounded. Sections1-4 are finite-regulator proofs; 000163 | no unjustified infinite trace or everywhere-defined graph chart is used. 000164 | Equation(5) as complement of an orthogonal projection remains a bounded 000165 | Hilbert-space identity wherever the original G is defined. 000166 | 000167 | The changed direct scalar graph-frame producer is therefore excluded in 000168 | its stated class, while the original whole completeness and nonlocal 000169 | source pairings are preserved. No boson field distribution, primitive 000170 | S_adm identity, physical spectrum, global RP failure or TFPT NoGo has 000171 | been derived by this test. 000172 | 000173 | 7. Actual normal-dependent ordered source: Plucker extension 000174 | ----------------------------------------------------------- 000175 | This extension applies to the original finite regulated, fixed-frame, 000176 | normal-dependent source, not to an added model. Let its whole charged 000177 | one-face generator obey 000178 | h(t)=h(t)*, 000179 | R'(t)=-h(t)R(t), R(0)=I. 000180 | The h(t) at different times need not commute. Finite evolution is 000181 | invertible. Jacobi's identity and trace cyclicity give 000182 | d/dt det R=det R Tr(R^-1 R') 000183 | =-Tr h(t) det R, 000184 | det R(t)=exp[-int_0^t Tr h(s)ds]>0. (6) 000185 | The trace of a Hermitian matrix is real. The initial determinant1 fixes 000186 | the sign continuously and uniquely; it never crosses zero. No ordering 000187 | of the h's has been removed from R itself. For integrable finite h the 000188 | same identity holds almost everywhere and integrates to (6). 000189 | 000190 | The normalized graph isometry for this generally NON-HERMITIAN R is 000191 | E_R=[I;R](I+R*R)^(-1/2). 000192 | Use the unique positive inverse square root, whose determinant is real 000193 | positive. With the same fixed ambient ordering (all first-face basis 000194 | vectors, then all second-face basis vectors), the normalized exterior 000195 | product Omega_R has coefficients 000196 | a_R=det(I+R*R)^(-1/2)>0, 000197 | b_R=det R det(I+R*R)^(-1/2)>0. 000198 | The second equality follows by selecting the lower n rows of the graph 000199 | matrix, giving det[R(I+R*R)^(-1/2)]. No additional relative wedge sign 000200 | appears in this fixed ordering. A global phase of Omega_R cancels from 000201 | its rank-one projector in any event. Its same ambient off-diagonal entry 000202 | is exactly 000203 | 000204 | =det R/det(I+R*R)>0. (7) 000205 | 000206 | Consequently NO nonzero positive scalar integration over these ACTUAL 000207 | whole ordered selfadjoint-generator histories can produce the identity 000208 | on the full ambient Plucker space from their unchanged rank-one 000209 | Plucker projectors. The positive entry(7) cannot integrate to zero. 000210 | This extends the section4 AMBIENT Plucker candidate exclusion to the 000211 | whole declared ordered source class, including all its boson paths, 000212 | without a static-background or commuting-generator assumption. 000213 | 000214 | The raw arbitrary-projector theorem of section2 is NOT extended: for 000215 | non-Hermitian R the raw graph off-diagonal block need not be a positive 000216 | Hermitian operator, so its proof does not apply. Likewise J_graph in 000217 | (4) does not generally represent the complement for non-Hermitian R. 000218 | The actual orthogonal complement still exists and has the explicit 000219 | normalized graph isometry 000220 | F_R=[-R*;I](I+RR*)^(-1/2), 000221 | F_R*E_R=0, F_R*F_R=I, 000222 | F_R F_R*=I-G_R. 000223 | Hence G_R+(I-G_R)=I remains exact and its positive normalized average 000224 | still holds for EVERY normalized nu. This complement identity is not 000225 | a new physical J action; the original oriented conjugation has its own 000226 | actual dictionary and is preserved below. It still fixes no boson law. 000227 | 000228 | [OMITTED: non-scientific conversation or execution metadata] 000229 | [OMITTED: non-scientific conversation or execution metadata] 000230 | [OMITTED: non-scientific conversation or execution metadata] 000231 | [OMITTED: non-scientific conversation or execution metadata] 000232 | [OMITTED: non-scientific conversation or execution metadata] 000233 | [OMITTED: non-scientific conversation or execution metadata] 000234 | [OMITTED: non-scientific conversation or execution metadata] 000235 | [OMITTED: non-scientific conversation or execution metadata] 000236 | [OMITTED: non-scientific conversation or execution metadata] 000237 | [OMITTED: non-scientific conversation or execution metadata] 000238 | [OMITTED: non-scientific conversation or execution metadata] 000239 | 000240 | As before, the kernel used here is the existing CONDITIONAL graph/CAR 000241 | source. Its assignment as a whole physical bosonic configuration frame 000242 | has not been derived by this result. Integrating dynamical bosons need 000243 | not leave a quasifree quantum state; no such claim is made. A signed, 000244 | complex, nonlocal dual or differently quotiented actual source is not 000245 | replaced by the positive scalar Plucker ansatz being tested. No global 000246 | TFPT/source/RP NoGo, primitive action identity or selected boson measure 000247 | is concluded. 000248 | 000249 | 8. Normalized projection average for arbitrary ordered graph families 000250 | ------------------------------------------------------------------ 000251 | For a genuinely given POSITIVE CLASSICAL probability nu over the actual 000252 | conditional graph projections, suppose P=int G_R dnu is itself an 000253 | orthogonal projection at a compatible finite regulator. Then 000254 | int ||G_R-P||_HS^2 dnu 000255 | =Tr P+Tr P-2Tr P^2=0. 000256 | Thus G_R=P almost everywhere. This elementary extremality calculation 000257 | uses only projections and normalization, so it applies also to the 000258 | non-Hermitian ordered R of section7. It is not an extension of section2 000259 | to nonnormalized arbitrary measures. 000260 | For an invertible graph its range determines R uniquely. A constant 000261 | normal-independent generator then follows by fixed-ell logarithm as in 000262 | section3. A fixed endpoint ordered R need NOT fix an entire time path 000263 | h(t); only support on its actual endpoint level set follows. 000264 | This lemma does not assert that the whole original graded integral is 000265 | a positive classical mixture of conditional CAR states, or that its 000266 | fermion marginal must remain a projection after boson integration. 000267 | Those would be additional physical identifications. The lemma is a 000268 | necessary check on a route making those explicit identifications, not 000269 | a constraint imposed on all original sources.